In Does conductance in the transmission line model represent a physical quantity? it came up that a non-real characteristic impedance just means the line has some loss. With real transmission lines having a little (but perhaps negligible) loss, we could expect all real transmission lines to have a characteristic impedance with at least a small imaginary component.
But is that really true? Let's say \$Z = Y = 1+1j\$. The characteristic impedance of this line is real:
$$ \sqrt{ 1+1j \over 1+1j } = 1 $$
And the attenuation constant is 1 [corrected by edit]:
$$ \operatorname{Re}\sqrt{(1+1j)(1+1j)} = 1 $$
Which to my understanding, means this line is lossless. But how can this be when it has both a non-zero conductance and resistance?
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